Topology 45 (2006) 171–208 www.elsevier.com/locate/top Conjugacy problem in groups of oriented geometrizable 3-manifolds Jean-Philippe Préauxa,b,∗ aEcole de l’air, CREA, 13661 Salon de Provence air, France bCentre de Mathématiques et d’informatique, Université de Provence, 39 rue F.Joliot-Curie, 13453 marseille cedex 13, France Received 10 June 2003; received in revised form 23 January 2005; accepted 20 June 2005 A la mémoire de mon frère Gilles Abstract The aim of this paper is to show that the fundamental group of an oriented 3-manifold which satisfies Thurston’s geometrization conjecture has a solvable conjugacy problem. In other words, for any such 3-manifold M, there exists an algorithm which can decide for any couple of elements u, v of 1(M) whether u and v are in the same conjugacy class of 1(M) or not. More topologically, the algorithm decides for any two loops in M, whether they are freely homotopic or not. ᭧ 2005 Elsevier Ltd. All rights reserved. Keywords: Conjugacy problem; Fundamental group; 3-manifold; Thurston geometrization conjecture; JSJ decomposition; Seifert fibered space; Hyperbolic 3-manifold; Graph of group; Fuchsian group; Word-hyperbolic group; Relatively hyperbolic group 0. Introduction Since the work of Dehn [4–6], the Dehn problems (more specifically the word problem and the con- jugacy problem) have taken on major importance in the developments of combinatorial group theory all along the XXth century. But while elements of this theory provide straightforward solutions in special ∗ Corresponding author at: Centre de Mathématiques et d’informatique, Université de Provence, 39 rue F.Joliot-Curie, 13453 marseille cedex 13, France. E-mail address: [email protected] URL: http://www.cmi.univ-mrs.fr/∼preaux 0040-9383/$ - see front matter ᭧ 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.top.2005.06.002 172 J.-P. Préaux / Topology 45 (2006) 171–208 cases, there still remain important classes of groups for which actual techniques fail to give a definitive answer. This is particularly true when one considers the conjugacy problem. This last problem seems to be much more complicated than the word problem, and even very simple cases are still open or admit a negative answer. Let us focus on fundamental groups of compact manifolds, a case which motivated the introduction of these problems by Dehn. Dehn has solved the word and conjugacy problems for groups of surfaces. It is well known that these two problems are in general insolvable in groups of n-manifolds for n4 (as a consequence of the facts that each finitely presented group is the fundamental group of some n-manifold for n4 fixed, and of the general insolvability of these two problems for an arbitrary f.p. group). In case of dimension 3, not all f.p. groups occur as fundamental groups of 3-manifolds, but the problems are still open for this class, despite many improvements. To solve the word problem, one needs to suppose a further condition, namely Thurston’s geometrization conjecture. In such a case, the automatic group theory gives a solution to the word problem, but fails to solve the conjugacy problem (cf. [8]). Some special cases nevertheless, admit, a solution. The small cancellation theory provides a solution for alternating links, and the biautomatic group theory provides solutions for hyperbolic manifolds and almost all Seifert fiber spaces; the best result we know applies to irreducible 3-manifolds with a non-empty boundary: they admit a locally CAT(0) metric and hence their groups have a solvable conjugacy problem [2]. A major improvement has been the solution of Sela [34] for knot groups. In his paper, Sela conjectures that “the method seems to apply to all 3-manifolds satisfying Thurston’s geometrization conjecture”. We have been inspired by his work, to show the main result of this paper: Main Theorem. The group of an oriented 3-manifold satisfying Thurston’s geometrization conjecture has a solvable conjugacy problem. Let us first give precise definitions of the concept involved. Let G =X|R be a group given by a finite presentation. The word problem consists in finding an algorithm which decides for any couple of words u, v on the generators, if u = v in G. The conjugacy problem consists in finding an algorithm which decides for any couple of words u, v on the generators, if u and v are conjugate in G (we shall write u ∼ v), that is, if there exists h ∈ G, such that u = h.v.h−1 in G. Such an algorithm is called a solution to the corresponding problem. It turns out that the existence of a solution to any of these problems does not depend upon the finite presentation of G involved. Novikov [26], has shown that a solution to the word problem does not exist in general. Since a solution to the conjugacy problem provides a solution to the word problem (to decide if u = v just decide whether uv−1 ∼ 1), the same conclusion applies to the conjugacy problem. Moreover, one can construct many examples of groups admitting a solution to the word problem, and no solution to the conjugacy problem (cf. [22]). When one restricts one self to the fundamental group of a manifold, solving the word problem is equivalent to deciding for any couple of based point loops, whether they are homotopic or not with base point fixed, while solving the conjugacy problem is equivalent to deciding whether two loops are freely homotopic. Bya3-manifold we mean a compact connected oriented manifold with boundary, of dimension 3. According to the Moise theorem [23], we may indifferently use PL, smooth, or topological locally smooth, structures on such a 3-manifold. A 3-manifold is said to satisfy Thurston’s geometrization property (we will say that the manifold is geometrizable) if it decomposes in its canonical topological decomposition—along essential spheres J.-P. Préaux / Topology 45 (2006) 171–208 173 (Kneser–Milnor), disks (Dehn lemma) and tori (Jaco–Shalen–Johanson)—into pieces whose interiors admit a riemanian metric complete and locally homogeneous. (In the following we shall speak improperly of a geometry on a 3-manifold rather than on its interior.) Thurston has conjectured that all 3-manifolds are geometrizable. This hypothesis is necessary to our work, because otherwise one can classify neither 3-manifolds nor their group at present. To solve the conjugacy problem, we will first use the classical topological decomposition, as well as the classification of geometrizable 3-manifolds, to reduce the conjugacy problem to the restricted case of closed irreducible 3-manifolds, which are either Haken (i.e. irreducible and containing a properly embedded 2-sided incompressible surface, cf. [16,17]), a Seifert fibered space [33,16], or modelled on SOL geometry (cf. [31]). That is, we show that if the group of any 3-manifold lying in such classes has a solvable conjugacy problem, then the same conclusion applies to any geometrizable 3-manifold (Lemma 1.4). The cases of Seifert fibered spaces and SOL geometry are rather easy, and we will only sketch solutions respectively in Sections 5.3 and 7; the interested reader can find detailed solutions in my PhD thesis [29]. The Haken case constitutes the essential difficulty, and will be treated in detail. We can further suppose that the manifold is not a torus bundle, because in such a case the manifold admits either a Seifert fibration or a geometric structure modelled on SOL geometry. As explained above, we solve the conjugacy problem in the group of a Haken closed manifold by essentially applying the strategy used by Sela to solve the case of knot groups. Let us now focus on the case of a Haken closed manifold M. The JSJ decomposition theorem asserts that there exists a family of essential tori W, unique up to ambient isotopy, such that if one cuts M along W, one obtains pieces which are either Seifert fibered spaces or do not contain an essential torus (we say the manifold is atoroidal). According to Thurston’s geometrization theorem [39], each atoroidal piece admits a hyperbolic structure with finite volume. This decomposition of M provides a decomposition of 1(M) as a fundamental group of a graph of groups, whose vertex groups are the 1 of the pieces obtained, and edge groups are free abelian of rank 2. We then establish a conjugacy theorem (in the spirit of the conjugacy the- orem for amalgams or HNN extensions), which characterizes conjugate elements in 1(M) (Theorem 3.1). This result, together with the algebraic interpretation of the lack of essential annuli in the pieces obtained (Proposition 4.1), and with a solution to the word problem, allows us to reduce the conjugacy problem in 1(M) into three algorithmic problems in the groups of the pieces obtained: the conjugacy problem, the boundary parallelism problem, and the 2-cosets problem (Theorem 4.1). Suppose M1 is a piece, and T is a 1 1 boundary subgroup of 1(M1) (that is, T =i∗(Z⊕Z) ⊂ 1(M1) for some embedding i : S ×S −→ jM1). The boundary parallelism problem in 1(M1) consists in finding for any element ∈ 1(M1) all the elements of T conjugate to in 1(M1). Suppose T1,T2 are two boundary subgroups (possibly identical); the 2-cosets problem consists in finding for any couples , ∈ 1(M1),all the elements c1 ∈ T1, c2 ∈ T2, such that = c1. .c2 in 1(M1). We then solve these algorithmic problems, separately, in the Seifert case, and in the hyperbolic case, providing a solution to the conjugacy problem in 1(M).